A Priori Error Bounds and Parameter Scalings for the Time Relaxation Reduced Order Model

Abstract

The a priori error analysis of reduced order models (ROMs) for fluids is relatively scarce. In this paper, we take a step in this direction and conduct numerical analysis of the recently introduced time relaxation ROM (TR-ROM), which uses spatial filtering to stabilize ROMs for convection-dominated flows. Specifically, we prove stability, an a priori error bound, and parameter scalings for the TR-ROM. Our numerical investigation shows that the theoretical convergence rate and the parameter scalings with respect to ROM dimension and filter radius are recovered numerically. In addition, the parameter scaling can be used to extrapolate the time relaxation parameter to other ROM dimensions and filter radii. Moreover, the parameter scaling with respect to filter radius is also observed in the predictive regime.

Figures (9)

• Figure 1: 2D flow past a cylinder at $\rm Re=100$, TR-ROM with $\chi=0.2$ and $\delta=0.04$. (a) Behavior of the mean squared $L^2$ error $\varepsilon_{L^2}$ with respect to $\Lambda^r_{L^2}$, and (b) behavior of the mean squared $H^1_0$ error $\varepsilon_{H^1_0}$ with respect to $\Lambda^r_{H^1_0}$.
• Figure 2: 2D flow past a cylinder at $\rm Re=100$. Behavior of $\chi_{\text{theory}}$ (\ref{['eq:opt_chi']}) and $\chi_{\text{effective}}$ with respect to the filter radius $\delta$ for $r=2~\text{and}~3$.
• Figure 3: 2D flow past a cylinder at $\rm Re=100$. Behavior of the extrapolated $\chi$ with respect to the filter radius, $\delta$, for $r=2~\text{and}~3$. Extrapolation is done by using the two values of $\chi_{\text{theory}}$ (\ref{['eq:opt_chi']}) and $\chi_{\text{effective}}$ at $\delta=0.2$ and $\delta=0.3$.
• Figure 4: 2D lid-driven cavity at $\rm Re=15,000$, TR-ROM with $\chi=0.05$ and $\delta=0.06$. (a) Behavior of the mean squared $L^2$ error $\varepsilon_{L^2}$ with respect to $\Lambda^r_{L^2}$, and (b) behavior of mean squared $H^1_0$ error $\varepsilon_{H^1_0}$ with respect to $\Lambda^r_{H^1_0}$. The ranges of $\Lambda^r_{L^2}$ and $\Lambda^r_{H^1_0}$ for the TR-ROM correspond to $r=8$ to $r=16$.
• Figure 5: 2D lid-driven cavity at $\rm Re=15,000$. Behavior of $\chi_{\text{theory}}$ (\ref{['eq:opt_chi']}) and $\chi_{\text{effective}}$ with respect to filter radius, $\delta$, for $r=4~\text{and}~16$.

Theorems & Definitions (21)

• Lemma 2.1
• Lemma 2.2: Discrete Gronwall Lemma heywood1990finite
• Remark 2.1
• Remark 2.2
• Definition 2.1: ROM $L^2$ Projection
• Lemma 2.3: $H_0^1$ POD Projection error
• proof
• Lemma 2.4: POD Inverse Estimates xie2018numerical
• Lemma 2.5: $L^2$ Stability of of ROM $L^2$ Projection xie2018numerical
• Lemma 2.6: Modified Lemma 3.3 in iliescu2014variational